this must be related to the old, mostly obsolete definition of prove that means test. "the proof of the pudding is in the eating" really means "the test of the pudding." It's still used in "proving grounds" & probably elsewhere, but that meaning isn't so apparent in English anymore. But it's a good way to remember it in norsk!
I understand that on it's own 'det' is translated as 'it' or 'that' and 'dette' as 'this', but I believe in sentences like this, it is quite common in English to use 'this'. What I mean is that in Norwegian 'det' seems to be much more common than 'dette'. In English on the other hand 'this' and 'that' seem to be much more interchangeable in the every day language. I would definitely say "You should try this" where 'this' would refer to whatever activity or thing was mentioned in the previous sentence(s).
"Å prøve" has two main definitions:
1: to test, try, control, investigate (alt. verbs: "å teste" (I, II), "å kontrollere"(III), "å undersøke" (IIII))
"Vi har [prøvd/testet] [/ut] det nye kurset."
"We've [tested/tried out] the new course."
"Har du [prøvd/smakt] fisken?"
"Have you [tried/tasted] the fish?"
2: to try, attempt, seek (alt. verbs: "å forsøke" (I, II), "å søke" (III))
"Vi har [prøvd/forsøkt] å fullføre det nye kurset."
"We've [tried/attempted] to finish the new course".
"Jeg skal [prøve/forsøke] å slå rekorden."
"I will [try/attempt] to beat the record."
"Å forsøke" only covers the second definition, while "å prøve" has even more definitions.